# APEX Calculus

Gregory Hartman, Virginia Military Institute

Brian Heinold, Mount St. Mary’s University

Troy Siemers, Virginia Military Institute

Dimplekumar Chalishajar, Virginia Military Institute

Jennifer Bowen, The College of Wooster

Copyright Year: 2014

ISBN 13: 9781514225158

Publisher: APEX Calculus

Language: English

## Conditions of Use

Attribution-NonCommercial

CC BY-NC

## Reviews

I have used this the semester-based courses Calculus 1 and Calculus 2. I will not comment on Calculus 3. The text addresses all of the required areas for the courses, though I would like to see a couple other areas: (1) a section providing... read more

I have used this the semester-based courses Calculus 1 and Calculus 2. I will not comment on Calculus 3. The text addresses all of the required areas for the courses, though I would like to see a couple other areas: (1) a section providing some of the motivation for the development of Calculus. Some information is provided in a couple paragraphs at the beginning of the Limits Chapter [Chapter 1], but no examples or exercises related to the motivation. In fact, the motivation for limit is not really addressed until Chapter 2. I supplement this on day 1 of my course. (2) While approximations with differentials is well-addressed, the related "Linear Approximation" topic is not. There is an example of a Linear Approximation in section 2.2 of the text, and several exercises related to it, but it is rarely addressed afterward. This is another topic I supplement. (3) Applications tend to be only addressed in specific application sections, rather than introduced early and called back upon after learning new rules or techniques. (4) Most of the conceptual exercises are either true/false, or open ended. I would like to see more conceptual exercises with graphical or numerical data. The text lacks proofs of most of the stated theorems. I am fine with that, and supplement the proofs that I think are most instructive. My classes see very few Mathematics majors and many Engineering majors, so I do not think that showing the proofs for all of the theorems is necessary, as most of my students will never need the proofs, but will need the appropriate techniques.

I have not found any errors in the writing, nor areas of bias.

I have not found any use of "current" data for examples or exercises, so there should not be any issues with the updating.

Text is readable, using notation appropriate for the level of students that are taking the course.

The terminology of the text is consistent throughout. I have not found any issues.

The text is well structured. Sections always start on new pages. Examples and Theorems have specific "call-outs" that are used consistently throughout the text. There are clear Sub-Headings within sections for sub-topics. Exercise sets always start on new pages.

Very good, overall. Though there are a couple areas that I have concerns with: (1) The section on Hyperbolic Functions is presented in Chapter 6 with Techniques for Antidifferentiation. The derivatives and antiderivatives are presented in the section, but I think I would have broken that up into two separate topics, and placed the derivative techniques in an earlier chapter. (2) L'Hopital's Rule is also presented in Chapter 6 with Techniques for Antidifferentiation, immediately before Improper Integration. I think that L'Hopital's Rule is useful for curve sketching, so I prefer to teach it as an application of differentiation.

I have not found any issues with the interface. The text has appropriate bookmarks that are all linked correctly. Graphs show correctly. The large-size version of the PDF has graphs that can be dragged to show different perspectives (useful for Calculus 3).

I have not found any grammatical issues.

There is nothing offensive in the course. But the examples do not show any inclusiveness.

Overall an excellent text, though there are some areas that I tend to supplement. If you are interested in getting into using OER textbooks for a Calculus course. This one would be a good choice to start with.

This a is a comprehensive text that covers all the basic material presented in a standard calculus sequence. It is clearly written, with easy-to-understand explanations. Many formal proofs are omitted, but the theorems and ideas are explained well. read more

This a is a comprehensive text that covers all the basic material presented in a standard calculus sequence. It is clearly written, with easy-to-understand explanations. Many formal proofs are omitted, but the theorems and ideas are explained well.

The text is accurate, error-free, and unbiased.

The content is up-to-date. It is relevant and in no way seems dated. The use of the "classic" position-velocity-acceleration example will remain applicable indefinitely.

The writing is clear and easy to understand. The explanations are concise and to-the-point. The examples are well chosen and show a variety of different situations and problem-solving techniques.

The text is consistent in its use of notation, symbols, terminology, and framework.

The section length is appropriate for the material presented and a typical class period. The text is well organized within each chapter and section.

The organization of the book is standard and presents the material in a logical and clear fashion.

The interface is user-friendly. The table of contents and search features works well. The type-face is easy to read, and the graphs and tables are clear.

The text contains no grammatical errors.

The text is not culturally insensitive or offensive in any way.

Overall, this is a concise, compact, straightforward, understandable, but complete approach to a calculus textbook. It doesn't include some of the "extras" that one would expect from a more traditional textbook such as a pre-calculus review, interesting application problems, larger projects for students to complete, or rigorous proofs of all the theorems. However, all the basic material is included.

The text covers all areas and skills of Calculus I, Calculus II, and Calculus III. This is an excellently written standard Calculus text that includes all ideas and skills of comprehensive college Calculus sequence. The text provides answers to... read more

The text covers all areas and skills of Calculus I, Calculus II, and Calculus III. This is an excellently written standard Calculus text that includes all ideas and skills of comprehensive college Calculus sequence. The text provides answers to exercises and an effective index.

The content is accurate, error-free, very-well written, professionally organized and unbiased.

The content is perfectly up-to-date. This comprehensive college Calculus textbook will not become obsolete. The text is professionally written and arranged in such a way as to make any future updates easy and straightforward to implement.

The text is written clearly and lucidly, with great variety of excellent examples and clear, concise explanations. It provides adequate context for mathematical terminology used in the text.

The text is effectively organized into a standard Calculus sequence. It is professionally written and arranged. The text is internally consistent in terms of terminology and framework.

The text is effectively organized into smaller sections and units that can be assigned at the appropriate points within the course. The text is professionally written, arranged, and properly formatted into easily manageable units well-suited for college students. The text can be easily reorganized and realigned with the requirements of any Calculus course without presenting any disruption to the reader.

The textbook is very effectively organized. The topics in the text are presented clearly and arranged logically.

The text is user-friendly. It is easy to navigate. All text images, graphs, and charts are clear. PDF file is up-to-date and easily navigable. Nothing to distract or confuse the reader..

The text contains no mathematical or grammatical errors.

This is an excellent Calculus textbook. The text is professionally written and edited. It is not culturally insensitive or offensive in any way. This very effective text is aimed at all college students regardless of race, ethnicity, creed, or background.

I have been using this book since 2015 for both lecture and online Calculus classes. My students are from various racial, ethnic, cultural, and religious backgrounds, as well as different countries of origin. Some students use an online version of the textbook while other purchase a printed copy. This book is well-liked by virtually all students. It is clear, excellently organized, and easy to use. Highly recommended.

The text covers material for a first semester course in differential calculus and begins integral calculus with antiderivatives and Riemann sums. The book begins with limits (even the epsilon-delta definition) and continuity before delving into... read more

The text covers material for a first semester course in differential calculus and begins integral calculus with antiderivatives and Riemann sums. The book begins with limits (even the epsilon-delta definition) and continuity before delving into derivatives and their applications (e.g. curve sketching, Newton's method, related rates, and optimization).

Content is accurate, error-free and unbiased.

Content is up-to-date and should remain so indefinitely. The non-mathematical application primarily used are those dealing with position, velocity, and acceleration.

The text has the appropriate amount of prose with adequate mathematically written explanations.

The text is internally consistent in terms of terminology and framework. The text terminology standard to most differential calculus books, such as product rule, quotient rule, and chain rule.

The section lengths within chapters are appropriate for a one hour class meeting, if the students have read and thought through the section beforehand.

The organization of the book is standard: limits and continuity, differentiation rules, curve sketching, applications. The book doesn't take too much time in connecting the ideas of limits and differentiation. Instructors should expect to spend more time explaining this connection.

The book is freely available as a PDF with hyperlinked table of contents. The third volume's (i.e. for multivariable calculus) PDF allows the user to manipulate the graphics.

The text contains no grammatical errors.

The text is not culturally insensitive or offensive in any way.

As the author points out in the preface, the number of exercises at the end of each section is not too large. Students can be expected to complete all of the exercises. The reader of this text can skip the "epsilon-delta" definition for limits without too much frustration.

The text covers all necessary topics for Calculus I and II. However, no justification or proof for the derivatives of the natural exponential and natural logarithmic functions are provided. They are simply stated among the basic rules for... read more

The text covers all necessary topics for Calculus I and II. However, no justification or proof for the derivatives of the natural exponential and natural logarithmic functions are provided. They are simply stated among the basic rules for differentiation without development. Most topics in Multivariable Calculus are included, with the exception of vector fields. No review of Precalculus topics is included, nor is any historical development of calculus. Conversational discussions of theorems are used in place of formal proofs in many cases throughout the entire text. An index is included, and the search function is accurate. If the text is to be used for Calculus I and II only, I would rate its overall comprehensiveness at 4-5, based on the comments above. For Multivariable Calculus, I would rate it at 3.

For the most part, the accuracy of the mathematical content is excellent. In a careful reading of the Calculus I material, a few typographical errors and one mathematical error (which might also have been typographical) were found. Since there is no discussion of people in the text, it contains no content that I would construe as biased.

There is no topic in the text which would make the material outdated, though a lack of interactive apps, web links, and computer-generated graphics may make it appear less stimulating than the modern for-profit, online textbook. Updates may be made difficult by the choice of numbering definitions, theorems and key ideas from 1 to n throughout the text, rather than by chapter and number (e.g. Thm. 2.5).

The text is extremely readable for the first-time calculus student. My notes repeatedly include the words "clear", "understandable", and "straightforward". The explanations of the concepts of the limit, the derivative, differentials, integration, sequences, and series are conversational, accurate, and lucid. Applications are well-explained, though in some cases (e.g. the disk and washer methods of determining volumes) more and better graphs and pictures would be appreciated.

Notation and terminology are consistent.

For the most part, chapter sections are divided as expected for a calculus text. Section 6.1 is remarkably long, including not only integration by substitution, but trigonometric integrals as well. Institutions that include only basic u-substitutions in Calculus I and trigonometric integrals in Calculus II will need to divide this section. As noted earlier, the text is quite readable, and the division of chapters into sections, and further into segments of concept development, and examples are appropriate.

The organization of the text is logical and consistent, and its flow is smooth. As previously noted, theorems are not generally justified with formal proofs but with conversational discussions. Topics are ordered appropriately, except, in my opinion, for the presentation of the derivatives of exponential and logarithmic functions without background development.

Navigating by use of the table of contents, bookmarks, or by page number is error-free. Highlighting and "sticky notes" are available. An index is included, but the search tool is quick and easy to use. Graphs are clear. No other pictures or images are included, and no internet links are included (which could become outdated). If students/instructors choose to print a hard copy of the text, there is blank space at the bottom of each page for hand-written notes.

The only grammatical errors found (notably, all in section 2.1) were likely typographical. Sentence structure, choice of words, and punctuation were all very good.

This item is not very applicable to the text, as no mention of culture or ethnicity is made. A check of several examples and application problems that refer to people appear to refer to women as often as they refer to men.

Problem sets are included at the end of each section, and answers to selected problems (most of the odd problems) are found at the end of the text. However, the sets of problems are generally more limited that what is found in a traditional textbook, and answers are sometimes spare. (For example, proofs are "left to the reader".) No review sections or problem sets are included at the end of chapters.

## Table of Contents

- Chapter 1: Limits
- Chapter 2: Derivatives
- Chapter 3: The Graphical Behavior of Functions
- Chapter 4: Applications of the Derivative
- Chapter 5: Integration
- Chapter 6: Techniques of Antidifferentiation
- Chapter 7: Applications of Integration
- Chapter 8: Sequences and Series
- Chapter 9: Curves in the Plane
- Chapter 10: Vectors
- Chapter 11: Vector Valued Functions
- Chapter 12: Functions of Several Variables
- Chapter 13: Multiple Integrations
- Chapter 14: Vector Analysis

## About the Book

This text comprises a three–text series on Calculus. The first part covers material taught in many “Calc 1” courses: limits, derivatives, and the basics of integration, found in Chapters 1 through 6.1. The second text covers material often taught in “Calc 2:” integration and its applications, along with an introduction to sequences, series and Taylor Polynomials, found in Chapters 5 through 8. The third text covers topics common in “Calc 3” or “multivariable calc:” parametric equations, polar coordinates, vector–valued functions, and functions of more than one variable, found in Chapters 9 through 14. More information, including free downloads of .pdf versions of the text, is available at www.apexcalculus.com.

## About the Contributors

### Authors

**Gregory Hartman**, PhD. Author. Associate Professor of Mathematics at Virginia Military Institute, where he has been on faculty since 2005. He earned his PhD in Mathematics from Virginia Tech in 2002.

**Brian Heinold**, PhD. Contributor. Associate Professor, Mathematics and Computer Science Department, Mount St. Mary's University. Heinold came to Mount St. Mary's in 2006, after receiving his doctorate from Lehigh University. Since then he has taught a variety of math and computer science courses. He has mentored several honors projects, coordinated the department's Smalltalk colloquium series and advised a number of COMAP teams. He has given presentations on fractals and mathematical imagery, teaching and graph theory.

**Troy Siemers**, PhD. Contributor. Head of the Applied Mathematics program at VMI. He earned his Ph. D. from the University of Virginia and previously led a summer program abroad in Lithuania.

**Dimplekumar Chalishajar**, PhD. Contributor. Assoicate Professor, Department of Applied Mathematics and Computer Science, Virginia Military Institute.

### Editor

**Jennifer Bowen**, PhD. Associate Professor and Department Chair of Mathematics and Computer Science, The College of Wooster. Bowen earned a BA in Mathematics with Honors from Boston College, and both an MS and PhD in Mathematics from The University of Virginia. Bowen teaches a range of courses, including Math in Contemporary Society, Basic Statistics, Calculus I, Calculus II, Multivariate Calculus, Transition to Advanced Mathematics, and Abstract Algebra.